GCSE Maths Practice: inverse-proportion

Inverse Proportion with Workers and Time

This question involves inverse proportion, a common GCSE Maths topic that often appears in problems about people, machines, or time. In inverse proportion, as one quantity decreases, the other must increase so that the overall result stays the same.

In work-related questions, the key idea is that the total amount of work does not change. Whether more people work for a shorter time or fewer people work for a longer time, the job itself remains identical.

The Key Principle

For questions involving workers and time, the following rule always applies:

number of workers × time taken = constant

This relationship helps you decide whether a question involves direct or inverse proportion. If fewer workers are doing the same job, the time must increase.

Step-by-Step Approach

1. Identify the two quantities involved (workers and time).
2. Calculate the total amount of work using the first situation.
3. Keep this total the same for the new situation.
4. Form a simple equation and solve it.

Writing these steps clearly is important in GCSE exams, as it shows correct mathematical reasoning.

Worked Example (Different Numbers)

Example: 4 workers can complete a task in 18 hours. How long would it take 6 workers to complete the same task?

• Total work = 4 × 18 = 72
• Let the new time be t
• 6 × t = 72
• t = 12 hours

This example shows how increasing the number of workers reduces the time needed.

Another Example Using Fewer Workers

Example: 8 people take 10 hours to complete a job. How long would it take 5 people?

• Total work = 8 × 10 = 80
• 5 × t = 80
• t = 16 hours

Here, the number of workers decreases, so the time increases.

Common Mistakes

• Using direct proportion instead of inverse proportion.
• Adding or subtracting values instead of multiplying.
• Forgetting that fewer workers means more time, not less.
• Not keeping the total work the same in both situations.

Real-Life Context

Inverse proportion appears in everyday situations. For example, if fewer people help clean a hall, it will take longer. In construction or manufacturing, reducing the workforce usually increases the time needed to finish a task.

Frequently Asked Questions

How do I recognise inverse proportion?
If one quantity goes down while the other goes up for the same task, it is likely inverse proportion.

Do I always multiply first?
Yes. In work problems, multiplying helps you find the total amount of work.

Study Tip

Before calculating, ask yourself: “If there are fewer workers, should the time be bigger or smaller?” This quick check helps you avoid simple exam mistakes.